We use neural networks as control variates with geometric integration techniques.
problem Analytic integration of neural network approximations for variance reduction.
method Integration domain subdivision using computational geometry for MLPs with continuous piecewise linear activation functions.
result Neural networks can be used as control variates with geometric integration methods.
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
The paper defines and proves the existence of decompositions of integral varifolds.
problem Existence of integral varifold decompositions.
method Introducing and proving the existence of decompositions of integral varifolds into countably many integral varifolds.
result Existence of decompositions of integral varifolds whose first variation is representable by integration.
We develop variational integrators from discrete Hamiltonian systems with external forces.
problem Creating accurate discrete models of continuous Hamiltonian systems.
method Constructing discrete Hamiltonian systems with external forces, analyzing symplectic structure, and combining methods to build variational integrators.
result We derive variational integrators that approximate continuous Hamiltonian systems with high accuracy.
Derives new optimization methods using variational integrators.
problem Optimization methods in machine learning.
method Variational integrators and principles of Hamilton and Lagrange-d'Alembert.
result Derives two families of optimization methods, including Nesterov's accelerated gradient method.
The paper studies the smoothness of critical points of variational integrals on Hessian spaces.
problem The study focuses on the regularity of critical points of variational integrals defined on Hessian spaces.
method The approach involves solving a fourth order nonlinear equation and analyzing the Hessian of the critical points.
result Smooth critical points with bounded Hessian are shown to be smooth provided their Hessian has small BMO.
Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of the first--order variational integrals related to the second--order Euler--Lagra…
This work introduces a fixed-point optimization for variational inference.
problem Improving quantified uncertainty in predictions by optimizing a simplified distribution over parameters.
method Projective integral updates for high-dimensional variational inference.
result Efficient quasirandom quadrature sequence for mean-field distributions, leading to quasi-Newton variational Bayes (QNVB).
A complete error analysis of variational integrators is obtained, by blowing up the discrete variational principles, all of which have a singularity at zero time-step. Divisions by the time step lead to an order that is one less than observed in simulations, a deficit that is repaired with the help of a new past-future…
We introduce the Variational Holder (VH) bound as an alternative to Variational Bayes (VB) for approximate Bayesian inference. Unlike VB which typically involves maximization of a non-convex lower bound with respect to the variational parameters, the VH bound involves minimization of a convex upper bound to the intract…
NCV uses neural networks to improve Monte Carlo integration.
problem Improving variance reduction in parametric Monte Carlo integration.
method NCV combines a normalizing flow and a neural network to approximate the integrand and solve the integral equation, with a neural importance sampler to estimate the difference.
result NCV achieves state-of-the-art performance in light transport simulation with reduced noise and negligible bias.
Geometric integrator preserves coadjoint orbits in dissipative systems.
problem Preserving coadjoint orbits in dissipative mechanical systems.
method Adapted discrete variational integrators for forced Euler-Poincaré and Lie-Poisson systems.
result Preserves coadjoint orbits exactly, improving over general-purpose methods.
New integral theorems improve density function estimations.
problem Improving density function estimations.
method Integrals based on cyclic functions and Riemann sums, Fourier integral theorem, Monte Carlo methods, variational approach, Cauchy residue theorem.
result Optimal cyclic functions minimize square integrals, improving density estimations.
Develops computational methods for simulating rigid body dynamics on SO(3).
problem Simulating rotational dynamics of rigid bodies on SO(3).
method Discrete Mechanics, Variational Integrators, Newton-Raphson algorithm.
result Preserves symplectic structure of SO(3) manifold dynamics.
Variational methods are employed in situations where exact Bayesian inference becomes intractable due to the difficulty in performing certain integrals. Typically, variational methods postulate a tractable posterior and formulate a lower bound on the desired integral to be approximated, e.g. marginal likelihood. The lo…
Study variation spaces for neural networks, linking them to approximation theory.
problem Understanding the variation spaces of shallow neural networks.
method Examined variation spaces defined by convex hulls and integral representations for a dictionary of functions.
result Found that Barron space, spectral Barron space, and Radon BV space are variation spaces for certain neural networks.
This work computes integral variation and monodromy maps for plane curve singularities.
problem Computing integral variation and monodromy maps for plane curve singularities.
method Constructing analytic models, vector fields, and gyrographs to compute maps explicitly.
result Effective algorithms and gyrographs for computing integral variation and monodromy maps.
New integrators for Lagrangian systems on homogeneous spaces derived from nonholonomic mechanics.
problem Numerical integration of Lagrangian systems on homogeneous spaces.
method Nonholonomic partitioned Runge-Kutta Munthe-Kaas (RKMK) methods on Lie groups.
result Preservation of properties in high-order numerical integrators.
Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of càdlàg functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation …
Infinitesimal variation of Action functional in classical (non-quantum) field theory with higher derivatives is presented in terms of well-defined intrinsic geometric objects independent of the particular field which varies. 'Integration by parts' procedure for this variation is then described in purely formal language…
In this article we study constrained variational problems in one independent variable defined on the space of integral curves of a Frenet system in a homogeneous space G/H. We prove that if the Lagrangian is G-invariant and coisotropic then the extremal curves can be found by quadratures. Our proof is constructive and …
The paper analyzes errors in mechanical systems with external forces.
problem Error analysis of mechanical systems with external forces.
method Analysis of variational integrators with contact order r for discrete mechanical systems. result The contact order of the integrator is the same as the contact order of the original systems.
Theory of space-time currents for geometric evolutions.
problem Analysis of geometric evolutions driven by dislocations.
method Development of space-time integral currents with bounded variation, introduction of Lipschitz deformation distance.
result Agreement of Lipschitz deformation distance with integral Whitney flat metric for boundaryless currents.
The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.
problem Approximating functionals of non-geometric rough paths.
method Extending rough paths with time and quadratic variation terms, proving uniform approximation.
result Linear functionals of extended signatures uniformly approximate continuous functionals.
In this paper we study quasi-linear system of partial differential equations which describes the existence of the polynomial in momenta first integral of the integrable geodesic flow on 2-torus. We proved in [3] that this is a semi-Hamiltonian system and we show here that the metric associated with the system is a metr…
New Holder bounds improve variational inference by flattening thermodynamic curves.
problem Improving variational inference by addressing performance gaps between theory and practice.
method Generalizing thermodynamic integration to weighted Holder mean, introducing Holder bounds.
result Holder bounds promise a one-step approximation of exact marginal log-likelihood.
Based on conservation laws for surface layer integrals for critical points of causal variational principles, it is shown how jet spaces can be endowed with an almost-complex structure. We analyze under which conditions the almost-complex structure can be integrated to a canonical complex structure. Combined with the sc…
Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.
problem Modeling compressible fluid dynamics with thermodynamic constraints.
method Variational discretization with discrete exterior calculus.
result Derives a nonholonomic variational integrator for NSF system.
In this paper we provide a variational derivation of the Euler-Poincaré equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others. Moreover, we study in detail the underlying geometry which is related to the notion of Poisson groupoid. Finally, we apply the…
Study connects surface classes to conservation laws.
problem Understanding CMC surfaces in space forms.
method Relates moment class to cohomology class, shows variational origin.
result Both classes have a variational origin as Noether currents.
Combines control variates and adaptive importance sampling for Monte Carlo integration.
problem Improving Monte Carlo integration accuracy with control variates and adaptive sampling.
method A quadrature rule combining control variates and adaptive importance sampling.
result Non-asymptotic bound on the probabilistic error of the procedure.
A theorem proves integrability of Fréchet tangent distributions.
problem Integrability of Fréchet tangent distributions on manifolds.
method Introduced Condition W, applied variational approach, used differential forms.
result Existence and uniqueness of maximal foliations.
Extends exterior diff. sys. to Lie algebroids with examples.
problem Invariant inverse problem of the calculus of variations
method Extends exterior differential systems to Lie algebroids, defines integral manifolds.
result Defines integral manifolds for exterior diff. systems on Lie algebroids.
Proves modular functors for SO(3) have integral Hodge structures.
problem Proving modular functors have integral Hodge structures.
method Based on homological models and geometric identification.
result Geometric construction of Hodge structures on SO(3) modular functors.
In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywh…
The paper defines and analyzes set-valued stochastic integrals for Lévy processes.
problem Defining and analyzing set-valued stochastic integrals for Lévy processes.
method Extending classical definitions to convoluted integrals with square-integrable kernels, and proving properties of set-valued convoluted stochastic integrals.
result Set-valued convoluted stochastic integrals can be explosive and take extended vector values.
InVA models image outcomes from multiple modalities, outperforming standard VAEs.
problem Understanding relationships across multiple imaging modalities in neuroimaging.
method Integrative Variational Autoencoder (InVA) framework for image-on-image regression.
result InVA accurately predicts PET scans from structural MRI, outperforming conventional models.
The paper explores variational principles for equations of maximal symmetry, providing new insights and results.
problem Exploring variational principles for equations of maximal symmetry.
method Study of variational and divergence symmetries for linear and nonlinear equations of maximal symmetry, providing first integrals in explicit form.
result Significantly different results and more general variational symmetry algebra for linear and nonlinear equations compared to previous studies.
Novel method combines physics priors for energy-conserving dynamics.
problem Learning long-term dynamics of complex physical systems from noisy data.
method Variational Integrator Graph Networks integrating energy constraint, high-order symplectic integrators, and graph neural networks.
result Improves predictive performance across single and many-body problems.
SVGP KAN integrates sparse variational GP with KANs for scalable probabilistic inference.
problem Lack of probabilistic outputs in standard KANs and cubic scaling of Gaussian Process methods.
method Sparse Variational GP-KAN combines KAN topology with sparse variational inference and permutation-based importance analysis.
result Enables probabilistic KANs to handle larger datasets with linear computational complexity.
The paper studies variations of σu-curvature for submanifolds in Riemannian manifolds.
problem Understanding the behavior of σu-curvature under variations of submanifolds. method Analyzes the functional of σu-curvature for submanifolds of arbitrary codimension in Riemannian manifolds. result Provides insights into the variational properties of σu-curvature. We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.
We provide a variational description of any Liouville (i.e. volume preserving) autonomous vector fields on a smooth manifold. This is obtained via a ``maximal degree'' variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide explicit for…
We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kähler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its…
New mass inequalities and proofs for causal variational principles.
problem Proving new mass inequalities for causal variational principles.
method Proved a new inequality for minimizers of causal variational principles and applied it to prove the positive mass theorem.
result Introduced a positive quasilocal mass and proved new mass inequalities.
ProJIVE integrates multiple data types to explain joint and individual variation.
problem Integrating multiple types of data on the same subjects.
method Probabilistic EM algorithm for JIVE framework.
result ProJIVE learns biologically meaningful courses of variation and improves accuracy.
Physics models integrated into VAEs improve generative performance and extrapolation.
problem Improving generative models with interpretability and robustness.
method Physics-based latent space in VAEs with regularized learning to balance physics and neural network components.
result Generative performance and extrapolation improvements demonstrated on synthetic and real-world datasets.